Open-Ended Math Questions Reveal Student Thinking

Careful, intentional, and mindful questioning is one of the most powerful tools a skillful teacher possesses (Costa & Kallick, 2000). Teachers can use open-ended questions during math instruction or assessments to learn how students are problem-solving.

A question is considered open-ended when it is framed in such a way that a variety of responses or approaches are possible (Small, 2009). As shown in Figure 1, open-ended math questions are designed to uncover student understanding and misunderstandings. The responses are used to inform instruction rather than to make evaluative decisions (Rose & Arline, 2009).

Teachers analyze students’ responses to questions in order to learn how they think. The responses reveal what students know and how they apply that knowledge. Teachers then use this information to design instruction that supports student learning. Additionally, open-ended questions provide opportunities for students to respond and contribute at their respective levels. This is especially important for struggling students, since they are likely to be passive learners (Lovin, Kyger, & Allsopp, 2004).

Figure 1illustrates how asking an open-ended question identifies students’ understanding along a continuum. Teachers use this information to determine where to begin new instruction.

Figure 1. Continuum of student responses and instructional decisions.

The examples below from Instructional Consultation and Assessment Team (ICAT) Manual Book 3 (Gravois, Gickling, & Rosenfield, 2011, pp. 82-84) illustrates examples of open-ended questions what the responses reveal about students’ understanding of a variety of math concepts.

Specific Assessment: Mathematical Thinking

Good Math Questions Share the Following Features

Begin with what the student knows

Engage students in the math skills and thinking that you are trying to assess

Require more than facts to resolve

Are open-ended and have more than one answer

Consider the following types of mathematical open-ended questions:

Number Sense

Jack has 12 pets. Some are dogs and some are cats. How many dogs could Jack have, and how many cats could Jack have?This question can show you:

Can the student count 12 items?

Does the student know how to begin?

Does the student start with 12 and count back, or does she start with a number and add on to 12? (i.e. 10 or 5)

Does the student need counters or can he solve it mentally?

Does the student demonstrate knowledge of an operational algorithm?

I made a two-color tower with 6 blocks. What might my tower look like?This question can show you: